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HRSB, 2008 ACT IX Cohort Sohael Abidi November 7 th, 2008 DAY 2

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HRSB, 2008 Differentiated Instruction When, How & Why we Differentiate within Mathematics Instruction?

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HRSB, 2008Differentiation Mathematics is no longer for the selected few. All children must be expected to succeed in mathematics. NCTM Curriculum Standards What’s Differentiated Instruction? - It is a chance to offer a variety of learning options that address different levels, interest and learning styles of diverse needs of students in mixed ability classrooms.

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HRSB, 2008 How will Differentiated Instruction change classroom procedures? When students are busy making up their own minds, the role of the teacher shifts. This new focus defines the teacher as one who is: circulating, redirecting, disciplining, questioning, assessing, guiding, directing, validating, facilitating, moving, monitoring, challenging, motivating, watching, moderating, diagnosing, trouble-shooting, observing, encouraging, suggesting, watching, modeling and clarifying. http://members.shaw.ca/priscillatheroux/teacherrole.html

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HRSB, 2008 The Truth… A student – centered, constructivist approach to teaching & learning can sometimes be: ▪ time consuming ▪ messy ▪ inefficient The benefits, however, can be…?

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HRSB, 2008 Benefits For the Student Every student has an opportunity to succeed; A single experience with success is enough for a student to approach new learning situations with confidence and motivation Opportunity is there to discover personal strengths and show multiple intelligences Less frustration due to confusion or boredom Benefits For the Teacher More sense of control over each student’s learning progress A greater understanding of each students ability to learn The reward of having a classroom that allows equal opportunity for success for all students

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HRSB, 2008 Differentiation Builds Upon… Prior knowledge Prior experiences Culturally defined values and norms Biological differences in cognitive development Home environment Maturity level Self-efficacy Culturally determined perceptions of school and learning and learninghttp://www.smcm.edu/academics/EdStudy/d7-Proj/Projects/ResearchSites/acbrowning/index.htm

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HRSB, 2008 Differentiated Planning Takes effort and practice Begin with the “Big Idea,” or “Enduring Questions” Plan learning experiences that aim to enhance understanding of the “Big Idea” The major concept for every child is the same The teacher plans several ways to approach learning the same concept The “Big Idea” serves as the anchor for the lesson Students travel on different paths of learning but end up at the same point with an understanding of the same major concept

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HRSB, 2008 A Key to Planning The Pre-Assessment (formative assessment) before the actual lesson planning Gathering information about what the students already know, and what they need to learn The Pre-Assessment paints a picture of the number of students who have developed concept mastery, who show some understanding, or who show a need for additional focus or instruction This information will help determine how many levels of a lesson need to be prepared, or how one could plan a lesson that is neither above nor below the capabilities of the students (see handout: “The Role of Assessment in a Differentiated Classroom”) http://www.scarsdaleschools.org/mathsymposium/RoleofAssessmentinaDifferentiatedClassroom_Handout.pdf

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HRSB, 2008 What we can Differentiate Content refers to the concepts and sub-concepts each student learns in a particular lesson. Process refers to the learning experiences that you choose to provide for your students to achieve an understanding of the content in a lesson. Products are the end result of the lesson. Each student applies what she or he has learned in the lesson to create a final product or to show their acquired skill

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HRSB, 2008 Strategies for Differentiating Content, Process & Product Adjusting Questions Compacting Curriculum Tiered Lessons Flexible Grouping Others: Acceleration/Deceleration Student Interest Peer Teaching Anchoring Activities Learning Profiles/Styles Buddy-Studies Independent StudyLearning Centers Projects Readiness / Ability Learning Contracts

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HRSB, 2008 Strategies to Differentiate: Adjusting Questions Adjusting the level of complexity of questions while considering: How does a student understand, define, or explain a task? How does a student organize their approach to the task? Does the student see relationships? Does the student relate this work to similar problems? Does the student vary their approach to different problems? Can they describe their strategy? Does the student show evidence of thinking ahead or backwards? Can the student generalize the process or results? Can the student self-evaluate? How does the student work in a group?

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HRSB, 2008 THE LOCKER PROBLEM A school has 1,000 lockers and 1,000 students. The students decide to have fun one day, so they take turns opening and closing the lockers, according to the following plan. The first student opens every locker. The first student opens every locker. The second student closes every second locker. The second student closes every second locker. The third student opens every third closed locker. The third student opens every third closed locker. The fourth student closes every fourth open locker. The fourth student closes every fourth open locker. The students continue in this manner until all 1,000 students have had their turn. The students continue in this manner until all 1,000 students have had their turn. When all the students are finished, how many lockers remain open?

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HRSB, 2008 Strategies to Differentiate: Curriculum Compacting When a teacher encounters a student who has already mastered a concept that other students have not. Decide on a level of a achievement that indicates “mastery” Pre-assess students to decide who has already “mastered” the material Plan enrichment activities or adjust content for learners who attain a mastery level Use a record to keep track of the progress of the students

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HRSB, 2008 Strategies to Differentiate: Tiered Lessons Content is presented at varying levels of complexity, but the process is the same for all students Lesson Tier according to: - students’ readiness (ability to understand a particular level of content) - learning profiles (style of learning) - interests (student interest in the topics to be studied) - Students work in teacher-assigned groups according to the chosen tiering strategy ( ex. Topic comprehension)

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HRSB, 2008 1. Using grid paper, draw a square that has side lengths of two units. Determine the perimeter and area. Try again with squares of different side lengths. Write about the relationship between the side length of a square and its perimeter and its area. (E4.1 page 5-81) 2. Explore the area and perimeter of squares and rectangles using the website below*. Make two observations about what you notice about changes in sizes of squares or changes in height or width of rectangles. Record one question you have about area or perimeter. 3. A farmer has 100m of fencing to make a pen for his pigs. He decides that a square or a rectangle would be the best shape. What are some possible sizes he could make? What would the area be of each shape? Which pen would you recommend? (E4.2 page 5-81) 4. Read the book, Spaghetti and Meatballs for All! (Burns, Marilyn. Spaghetti and Meatballs for All! Brainy Day Books, 1995) Retell the story in your own words to a classmate using tiles to illustrate the main ideas. 5. Make a display to show when area and perimeter are used in the real world. You may cut out pictures or draw them yourself or make a graphic organizer using words or pictures. 6. Use string or masking tape to define a rectangle on the floor with an area large enough for four students to sit comfortably during silent reading. Record the perimeter of the rectangle. Area and Perimeter

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HRSB, 2008 Strategies to Differentiate: Flexible Groupings Groupings can be decided based upon: - student interest; student readiness; student request Allow students to decide to leave a particular group if appropriate. (students may start at a slow pace, but progress into the group with less guidance.) Avoid labeling within a classroom Accommodate differences within an individual; Are always changing; students should not be able to predict what group they will be assigned to

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HRSB, 2008 Differentiating from our Textbook Mathematics 9: Focus on Understanding Refer to your differentiating handouts With a partner, complete each question. Differentiate each question by creating 3 versions of each question Create one version for students at three different ability levels ability levels Share with your table Group share

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HRSB, 2008 Looking at the TR CD Refer to Differentiation From Textbook handouts With a partner, look at the resource handouts printed from the TR CD Match the differentiated handouts to their corresponding textbook question Discuss the differentiation evident for each question (type, pros, cons, changes etc.) Group TR CD surf!

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HRSB, 2008 Effective Questioning: Raising the Cognitive Demand

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HRSB, 2008 What Role does Questioning play in our daily lessons?

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HRSB, 2008 Effective Questioning: Helps control the flow of information Helps control the flow of information Keeps students focused on important mathematical ideas Keeps students focused on important mathematical ideas Helps students make sense of mathematics Helps students make sense of mathematics Moves discussions from discrete, unrelated responses to in-depth dialogue Moves discussions from discrete, unrelated responses to in-depth dialogue Supports and encourages student thinking Supports and encourages student thinking * Ultimately, effective questioning helps to raise the ‘cognitive demand’ placed on our students during our daily lessons

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HRSB, 2008 Activity: “The Ways I Ask for Information” Questioning handout 1 Questioning handout 1 Write questions and purposes independently Write questions and purposes independently Complete & share with a partner Complete & share with a partner Group sharing session Group sharing session

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HRSB, 2008 Discussion: What is the Purpose? One can categorize questions in many ways, but an important part of questioning is: One can categorize questions in many ways, but an important part of questioning is: The Teacher’s Purpose for asking the questions That is, what is the teacher trying to accomplish? That is, what is the teacher trying to accomplish? In partners, think about a specific problem where questions are designed merely to get answers ‘on the table.’ In partners, think about a specific problem where questions are designed merely to get answers ‘on the table.’ Jot down the example and some questions that may be asked. Jot down the example and some questions that may be asked. Now create questions requiring higher level thinking for your example…raise the cognitive demand of your questions. Now create questions requiring higher level thinking for your example…raise the cognitive demand of your questions.

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HRSB, 2008 Classifying Questions based on Purpose Engaging: invite students into a discussion; keep them engaged in conversation; invite them to share their work, or get answers on the table Refocusing: help students get back on track or move away from a dead-end strategy away from a dead-end strategy Clarifying: help students explain their thinking or help you understand their thinking help you understand their thinking

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HRSB, 2008 Discussion Thoughts? Is there a ‘grey-area?’ Thoughts? Is there a ‘grey-area?’ In partners, describe an instructional situation that you have experienced and give an example of a question from each category. In partners, describe an instructional situation that you have experienced and give an example of a question from each category. Would an observer in your class know what your purpose is for each of these questions? Would an observer in your class know what your purpose is for each of these questions?

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HRSB, 2008Examples:Engaging “What strategies might we use to solve this problem?” “If you wanted to graph this function, how would you label the axis?” Refocusing – ( if students are stuck working on): Similar Figures – “ What does it mean for two figures to be similar?” to be similar?” Ratio – “What quantities are you comparing?” Ratio – “What quantities are you comparing?”Clarifying “How did you figure out your answer?” “Why did you start with that number?”

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HRSB, 2008 Task Revisit your “Asking for Information” sheet Revisit your “Asking for Information” sheet Determine the purpose for asking each of your written questions Determine the purpose for asking each of your written questions ex. Engaging, Refocusing, Clarifying

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HRSB, 2008 Discussion Volunteers: Read one of your questions, identify its category, and explain why it is the correct category Volunteers: Read one of your questions, identify its category, and explain why it is the correct category Comments?? Comments?? Which category was most popular based on our examples? Least popular? Why might this be? Which category was most popular based on our examples? Least popular? Why might this be? Could one question fall into more than one category? Let’s discuss! Could one question fall into more than one category? Let’s discuss!

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HRSB, 2008 Fraction Tracks 0 1 0 0 0 0 0 0 1 1 1 1 1 1 DEMO: 5/6

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HRSB, 2008 Time to Play! Fraction Tracks Handout Fraction Tracks Handout With a partner, alternating turns selecting fraction cards, try and be the first person to move all 7 chips from 0 to 1 on their respective number lines. With a partner, alternating turns selecting fraction cards, try and be the first person to move all 7 chips from 0 to 1 on their respective number lines. GO! GO!

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HRSB, 2008 ‘Post-Game Summary’ Sample Student Card & Comment: “ I drew a card with the fraction 8/8 on it. I don’t know what to do since I can’t use 8/8 on either the fourths or the eighths track.”

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HRSB, 2008 Below are a series of questions that you may have asked this student. Read the student answers and decide whether or not the question was effective. Think of another question that could follow up each student answer. Below are a series of questions that you may have asked this student. Read the student answers and decide whether or not the question was effective. Think of another question that could follow up each student answer. Teacher: Why can't you use the entire 8/8 on either of the tracks? Student : I can't use 8/8 on the fourths track because there's already a marker on 1/4, and I can't use 8/8 on the eighths track because there's already a marker on 2/8. Teacher : Can you use part of the 8/8 on either track? Student: Yes. Teacher: How much could you use on the fourths track? Student: The marker's at 1/4, so I could use 3/4.

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HRSB, 2008 Teacher : What is 3/4 equivalent to on the eighths track? Student: It's also 6/8, so I could use 6/8 on the eighths track.

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HRSB, 2008 Discussion & Debrief: Video Task Watch the Fraction Tracks video: (1:16-11:16) Watch the Fraction Tracks video: (1:16-11:16)Fraction Tracks Fraction Tracks http://www.learner.org/resources/series31.html Using the “Case Study” handout: Using the “Case Study” handout: - jot down any questions that are asked by the teacher; - include what you believe was the intended purpose of each question (engaging, refocusing, clarifying) each question (engaging, refocusing, clarifying) - after the video, share with a partner - group share - Discussion: volunteers to identify questions, and how/why they classified the purposes of these questions they classified the purposes of these questions

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HRSB, 2008 Post-Video Wrap-up Do the students seem to understand the mathematical ideas? Do the students seem to understand the mathematical ideas? Does the teacher ask questions that elicit, extend, and challenge the students' thinking? Does the teacher ask questions that elicit, extend, and challenge the students' thinking? What other questions could the teacher have asked this student in order to further challenge her mathematical thinking? What other questions could the teacher have asked this student in order to further challenge her mathematical thinking? Do you think the teacher's questioning of the students will help other students in the class develop their understanding of fractions? Do you think the teacher's questioning of the students will help other students in the class develop their understanding of fractions? Do you think the teacher listens, responds, and adapts her questions effectively? Do you think the teacher listens, responds, and adapts her questions effectively? What does the teacher do when a student gives an incorrect answer? What does the teacher do when a student gives an incorrect answer? Do you think the students understand the final answer, or are they just following the teacher's prompting without understanding? Do you think the students understand the final answer, or are they just following the teacher's prompting without understanding? Do you think this is an effective activity through which students learn important mathematical ideas? Do you think this is an effective activity through which students learn important mathematical ideas? How could you modify or extend this activity to teach other fraction concepts? How could you modify or extend this activity to teach other fraction concepts?

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HRSB, 2008 When Literacy Meets Math… Complete the griney grollers activity Complete the griney grollers activity (see handout) (see handout) Please after question #7. Please after question #7. STOP

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HRSB, 2008Discussion What types of questions were found in #1- 7? What types of questions were found in #1- 7? What was their purpose? (if any) What was their purpose? (if any) How meaningful were they in assessing student understanding and/or learning? How meaningful were they in assessing student understanding and/or learning? Now, continue with #8 thinking about how we can ask questions that increase the cognitive demand on our students. Now, continue with #8 thinking about how we can ask questions that increase the cognitive demand on our students. Using this example, what types of questions could we ask that would promote higher levels of student thinking and understanding? Using this example, what types of questions could we ask that would promote higher levels of student thinking and understanding?

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HRSB, 2008Handouts I stress the importance of having a “Question Toolbox” I stress the importance of having a “Question Toolbox” “Supporting Teachers in Asking Questions and Choosing Tasks” “Supporting Teachers in Asking Questions and Choosing Tasks” “Developing Mathematical Thinking with Effective Questions” “Developing Mathematical Thinking with Effective Questions” “Questions that Probe Understanding” “Questions that Probe Understanding” “Motivating Every Student Through Effective Questioning” “Motivating Every Student Through Effective Questioning” - “Summary of Questioning Techniques” (review with the group) (review with the group)

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HRSB, 2008 Higher Level Questioning within our daily activities Refer to the “Examples of Good Questioning” handout Refer to the “Examples of Good Questioning” handout Please complete number one, (top of sheet), on your own. Please complete number one, (top of sheet), on your own. Share strategies with a partner and discuss the questions that were asked (i.e. purpose, intension, level of required student thinking etc.) Share strategies with a partner and discuss the questions that were asked (i.e. purpose, intension, level of required student thinking etc.) If time, do the same for question two (bottom of sheet) If time, do the same for question two (bottom of sheet) The Cognitive Demand!

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HRSB, 2008 Scatter plots – Line of Best Fit Regression! A graph of ordered pairs of numeric data A graph of ordered pairs of numeric data Used to see relationships between two variables or quantities Used to see relationships between two variables or quantities Helps determine the correlation between the Independent & dependent variables Helps determine the correlation between the Independent & dependent variables Correlation: a measure of how closely the points on a scatter plot fit a line Correlation: a measure of how closely the points on a scatter plot fit a line The relationship can be strong, weak, positive or negative The relationship can be strong, weak, positive or negative + Correlation – As indep.Var ↑, Dep. Var ↑ + Correlation – As indep.Var ↑, Dep. Var ↑ - Correlation – As indep. Var ↑, Dep. Var ↓ - Correlation – As indep. Var ↑, Dep. Var ↓

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HRSB, 2008 Line of Best Fit Drawn through as many data points as possible Drawn through as many data points as possible Aim to have an equal amount of data points above and below the line Aim to have an equal amount of data points above and below the line Does NOT have to go through the origin Does NOT have to go through the origin Allows us to generate an equation that describes the relationship using an equation form (ie: y = mx+b) Allows us to generate an equation that describes the relationship using an equation form (ie: y = mx+b) Example 1, Pink Sheet 1 – Discuss (draw LOBF for each) Example 2, Pink Sheet 1, Let’s do together using the TI-83+ TI-83+

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HRSB, 2008 Calculator Applications: 10. (pg. 38-42 Booklet) Example 2: Line of Best Fit Example 2: Line of Best Fit 1. 2. 3. 4. 5. 6. 7. 8. 9.

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HRSB, 2008 Linear Regression & Correlation Coefficient (r) Determining the Equation for the Line of best fit can be referred to as: Regression Analysis Determining the Equation for the Line of best fit can be referred to as: Regression Analysis We create a model that can be used to predict values of the Dep. Var. based on values of the Indep. Var. We create a model that can be used to predict values of the Dep. Var. based on values of the Indep. Var. The ‘r’ value – Correlation Coefficient The ‘r’ value – Correlation Coefficient - measures the strength of the association of the 2 variables; (-1 → +1) – the closer to either, the stronger the relationship (-1 → +1) – the closer to either, the stronger the relationship Pink Sheet 3 – complete in table groups – (steps on page 4, 5 pink sheets) (steps on page 4, 5 pink sheets)

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HRSB, 2008 Regression Analysis Pg.383, Gr. 9 Text, #13 Window Scatter plotCorrelation EquationGraph EquationGraph

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HRSB, 2008 Extrapolating data: Determining # injured in 2010: Determining # injured in 2010: Change ‘window’ to include this x parameter (Xmax – 2050) The new graph: Next Key Strokes: 2 nd CALC 1:value Type in 2010 Y value when x = 2010, is

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HRSB, 2008 Regression Analysis Cont. Example 3, 4: Pink Sheet 3 - EXTENSION Example 3, 4: Pink Sheet 3 - EXTENSION - Looking at Parabolic & Exponential Relationships - Complete these problems together

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HRSB, 2008 Algebra Tiles 1 x -x x2x2 -x 2

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HRSB, 2008 Algebra Tiles y-y xy -y 2 y2y2 -xy

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HRSB, 2008 Question 1

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HRSB, 2008 Display first polynomial

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HRSB, 2008 Display first polynomial Add to your display the second polynomial

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HRSB, 2008 Simplify by combining like terms

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HRSB, 2008 Remove “zeros”

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HRSB, 2008 What remains is the answer!

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HRSB, 2008 Question 2

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HRSB, 2008 Display first polynomial expression

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HRSB, 2008 Remove second polynomial expression

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HRSB, 2008 What is left?

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HRSB, 2008 Question 3

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HRSB, 2008 Display first polynomial expression

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HRSB, 2008 Can we remove the second? We cannot remove an if is not in the original display…

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HRSB, 2008 We have to insert into the display without changing the value... Add in zero... (made of a positive and a negative )

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HRSB, 2008 Now we can remove the second polynomial

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HRSB, 2008 What remains is the answer!

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HRSB, 2008 Question 4

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HRSB, 2008 Display first polynomial Can you remove the second?

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HRSB, 2008 We cannot remove -3 Add in -3 and +3 (they make zero and will not change the value)

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HRSB, 2008 Now we can remove all of the second polynomial Your answer is what remains

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HRSB, 2008 Question 5

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HRSB, 2008 Display the first expression Can you remove the second expression?

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HRSB, 2008 Yes! Go ahead and remove the second expression The answer is what remains!

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HRSB, 2008 Question 6

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HRSB, 2008 Display the first expression Can we remove the second expression? Not the way it is!

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HRSB, 2008 We need to add in every term from the second expression. Remember to pair these terms with their opposite so the value of the expression does not change. Double Check that you have not changed the value of the original expression...

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HRSB, 2008 Now remove the second expression Your answer is what remains

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HRSB, 2008 Question 7

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HRSB, 2008 Display the first expression Can we remove the second? Not the way it is!

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HRSB, 2008 Add in the tiles needed. Make sure the value of the expression has not changed

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HRSB, 2008 Remove the second expression. What remains is the answer.

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HRSB, 2008 Question 8

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HRSB, 2008 Display the first expression Add to your display the terms of the second expression Combine like terms

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HRSB, 2008 Remove “zeros” What remains is the answer

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HRSB, 2008 Question 9

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HRSB, 2008 Display first expression Add the second expression to your display

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HRSB, 2008 Combine like terms Remove any zeros What remains is the answer

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HRSB, 2008 Question 10

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HRSB, 2008 Display first expression See if all of the terms from the second expression can be removed Not the way it is!

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HRSB, 2008 Insert zeros as needed

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HRSB, 2008 Remove second expression What remains is the answer!

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HRSB, 2008 How can we simplify the following expression?

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HRSB, 2008

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Can you arrange the tile display to create a rectangle? What are the dimensions of the rectangle or area model you created?

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HRSB, 2008 What are the dimensions of the rectangle you created? The dimensions are: 3

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HRSB, 2008 Example 2

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HRSB, 2008 Remember repeated addition means displaying the polynomial as many times as indicated by the scalar

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HRSB, 2008

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Can you arrange the tile display to form a rectangle? Notice that the dimensions are the factors of the original question

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HRSB, 2008 Example 3 Use repeated addition to simplify the following expression

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HRSB, 2008

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Rearrange to combine like terms

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HRSB, 2008 State your answer

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HRSB, 2008 Can you rearrange your tile display to make a rectangle? 3

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HRSB, 2008 Example 4 Make an area model to answer the next question

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HRSB, 2008 Example 4 Use the two factors to create the length and width of a rectangle.

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HRSB, 2008 4 Then fill it in to make a complete rectangle

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HRSB, 2008 4 What is the area?

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HRSB, 2008 What do you see?

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HRSB, 2008 Number 1 What are the dimensions of the rectangle? What is the area? What can you say about the partial products?

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HRSB, 2008 Number 2

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HRSB, 2008 Number 3

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HRSB, 2008 Number 4

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HRSB, 2008 Number 5

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HRSB, 2008 Number 6

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